Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30,34,35)(31,38,36,32)(33,37,41,43)(39,42,44,40);(5,7,10,8)(6,11,12,9)(15,21)(19,22)(23,27)(26,28)(30,35)(31,36)(33,41)(40,42);(6,12)(7,8)(13,15,18,21)(14,19,16,22)(17,23,25,27)(20,26,24,28)(29,31,34,36)(30,32,35,38)(33,39,41,44)(37,40,43,42);(1,2)(3,4)(5,8,10,7)(6,11,12,9)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32)(30,36)(31,35)(33,42)(34,38)(37,39)(40,41)(43,44);(5,9,10,11)(6,7,12,8)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33)(30,37)(31,39)(32,40)(34,41)(35,43)(36,44)(38,42)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fgh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5527010);